The interlaced-pair conjecture for ghost distributions

Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair, let $\mathfrak{a}$ be a chosen Cartan subspace, and let $\overline{\Delta}$ be its reduced root system with restricted Weyl vector $\overline{\rho}$. Set $\overline{\Delta}_{ev}=\overline{\Delta_{\overline{0}}}\setminus\{0\}$ and $\overline{\Delta}_{odd}=\overline{\Delta}\setminus\overline{\Delta}_{ev}$. For $\alpha\in\overline{\Delta}_{ev}$, let $r_{\alpha}$ be the corresponding reflection, and let $W$ be the subgroup of $GL(\mathfrak{a})$ generated by these reflections. For $p\in S(\mathfrak{a})$ and $w\in W$, define $(w.p)(\lambda)=p(w(\lambda+\overline{\rho})-\overline{\rho})$. The interlaced-pair conjecture. If $(\mathfrak{g},\mathfrak{k})$ is interlaced, then $HC(\mathcal{A}_{(\mathfrak{g},\mathfrak{k})})$ is the set of $p\in S(\mathfrak{a})$ such that: for every $\alpha\in\overline{\Delta}_{ev}$, with $n_{\alpha}=\frac{1}{2}\operatorname{dim}(\mathfrak{g}_{\alpha})_{\overline{1}}$, $(r_{\alpha}.p)=(-1)^{n_{\alpha}}p$; and, for every $\alpha\in\overline{\Delta}_{odd}$ and every $\lambda$ satisfying $(\lambda+\overline{\rho},\alpha)=0$,

$$p(\lambda+r\alpha)=(-1)^r p(\lambda-r\alpha)$$

for $1\leq r\leq\frac{1}{2}\operatorname{dim}(\mathfrak{g}_{\alpha})_{\overline{1}}$. This gives a conjectural description of the Harish–Chandra image for interlaced pairs; the source does not provide evidence that it has been proved or refuted.